The rhombic dodecahedron and semisimple actions of Aut(Fn) on CAT(0) spaces

Abstract

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT(0) spaces. If n ≥ 4 then each of the Nielsen generators of Aut(Fn) has a fied point. If n = 3 then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated Z4 ⊂ Aut(F3) leaves invariant an isometrically embedded copy of Euclidean 3-space E3 → X on which it acts as a discrete group of translations with the rhombic dodecahedron as a Dirichlet domain. An abundance of actions of the second kind is described. Constraints on maps from Aut(Fn) to mapping class groups and linear groups are obtained. If n ≥ 2 then neither Aut(Fn) nor Out(F n) is the fundamental group of a compact Kähler manifold. © Instytut Matematyczny PAN, 2011.